There seem to be a lot of suggestions on how to prop up an old or
failing septic system out there, but I don't know if there's any true
answer other than to contact the town/county engineer and get a new
system designed and installed. If you absolutely HAVE to work with
this system right now (believe me, I know how that goes - our new
septic cost $26,000), here's my cheap solution.

If you own the hill that is dumping all of the water into your leach
field, maybe diverting the runoff around your leach field would help.
Burying some drainage pipe perpendicular across the hill in a couple of
spots and running those into more drainage pipe running down the hill
but outside the perimeter of your septic field may divert enough of the
water to let the system work properly. You MIGHT need to check with
the local engineer to see if this is acceptable or not and to find out
how far from the leach lines you need to stay with your drainage pipe
and where you can discharge it.